Star-shaped periodic solutions for \(\dot x(t)=-\alpha\{ 1-\|x(t)\|^2\}R(\theta)x([t])\) (Q1599963)

The authors consider the two-dimensional differential equation with piecewise constant argument \[ \dot x(t)=-\alpha\{ 1-\|x(t)\|^2\}R(\theta)x([t]), \] where \([\cdot]\) means the greatest-integer function, \(R(\theta)\) is a rotation matrix and \(\alpha >0.\) The existence of a crucial value \(\alpha_0\) of the parameter \(\alpha\) is proved such that: a) if \(0<\alpha\leq \alpha_0\) there is not any nonconstant periodic solution; b) if \(\alpha > \alpha_0\) and \(\frac{\theta}{\pi}\) is rational, then the system has a star-shaped periodic solution; c) if \(\alpha > \alpha_0\) and \(\frac{\theta}{\pi}\) is irrational, then each solution orbit with \(\|x(t_0)\|<1\) densely fills out an annular region centered at the origin.